In order to solve the plane unsteady and quasi-static coupled thermoelasticity problems with concave convex boundary shape, the element free Galerkin method (EFG) is used to solve the problem, the Moving least square method（MLS） is used to construct the shape function, the Lagrange multiplier method is used to deal with the essential boundary conditions (the first kind of boundary conditions), the Voronoi adjacency criterion and the posteriori error formula are introduced to adaptively optimize the subsequent results; A new EFG adaptive model for unsteady quasi-static and coupled thermoelasticity problems is constructed. The temperature field and displacement field distribution in the planes with smooth and concave convex boundary shape are calculated under two-dimensional mixed boundary conditions, respectively. And the results are compared with those of finite element method. The difference between the results of finite element method and meshless method is characterized, and the effectiveness and accuracy of EFG for unsteady quasi-static thermoelasticity coupled problem are verified